Question

A truck is to be driven 200 km along a level highway at x km per hour. Petrol costs 8 cents per litre and is used at the rate of \[25+\frac{ x^2 }{ 112 }\] litres per hour. The driver receives 2 dollars per hour. What is the most economical speed (to the nearest km per hour and the cost of the trip-
i) if there is no speed limit,
ii) if the speed must not exceed 60km per hour?

Answer 1

Topic: Calculus
Sub Topic: Maxima and Minima.

Answer 2

Is it 74.83 km/h?

Answer 3

Yes!!

Answer 4

\[C(x) = 0.08(25 + \frac{ x^{2} }{ 112 }) * \frac{ 200 }{ x } + 2(\frac{ 200 }{ x })\]

This is the equation I came up with for Cost.

Answer 5

75km/hour

Answer 6

How did you do the first part?

Answer 7

sorry

Answer 8

gallon?

Answer 9

American?

Answer 10

let me rewrite that , (i used american units and accidentally wrote it in the wrong place)

Answer 11

The first thing we know is that the rate of petrol is $0.08/litre

And the second thing we know is that the petrol is used at a rate of 25 + x^2/112 litre/h

so

\[0.08 \frac{ dollar}{ litre } * (25 + \frac{ x^{2} }{ 112 }) \frac{ litre }{ hour }\]

Gives us the units dollars/hour

Answer 12

The only thing we need to know is how long the trip takes.

We know speed = distance / time

speed = x
distance = 200 km
time = t

Therefore time, t = 200/x

If we multiply what we have so far by (200/x) (which is in hours), then the hours unit cancels leaving an expression in dollars

Answer 13

So far, I'm up to:
\[0.08*(25+\frac{ x^2 }{ 112 })\]

Answer 14

\[0.08 \frac{ dollar }{ litre } * (25 + \frac{ x^{2} }{ 112 }) \frac{ litre }{ hour } * \frac{ 200 }{ x } hours\]

Answer 15

So you get that, and do you then add:
\[\frac{ 400 }{ x }\] for the money he receives per hour?

Answer 16

yes

Answer 17

okay. Now I'm getting somewhere.

Answer 18

So do we differentiate that with respect to x?

Answer 19

Yes because C(x) is the cost evaluated at x. So we need to minimize cost by differentiating with respect to x.

Answer 20

Thanks for the assistance @LogicalApple . Appreciate it.

Answer 21

Not a problem

Answer 22

For the second part of the question, keep in mind that the point we evaluated in the first part was a minimum.

Due to the nature of the hyperbola, you can consider it an absolute minimum whenever x > 0

This should help you answer the second part of the question.

Answer 23

Yeah I already finished the question.

Answer 24

Oh, nice

Answer 25

The hard part was what you did for me.

Answer 26

I did it on paper and I was like "hm.. I can't think in litres/gallon"

Answer 27

Ah, I did it again lol
I mean litres/hour!

Answer 28

hahah.

Answer 29

Well I'm going to close this and put up another question. Thanks again for helping me.

Answer 30

Hope you can help me again if you see fit to.

Answer 31

Good luck!